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# Modular arithmetic

In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" upon reaching a certain value—the modulus (plural moduli). 

122 relations: Abstract algebra, Addition, Advanced Encryption Standard, Apportionment (politics), Arithmetic, Binary relation, Bitwise operation, Boolean ring, Calculator, Carl Friedrich Gauss, Carmichael function, CAS Registry Number, Casting out nines, Characteristic (algebra), Charles E. Leiserson, Check digit, Chemistry, Chinese remainder theorem, Circular buffer, Clifford Stein, Commutative ring, Computer algebra, Computer science, Congruence relation, Coprime integers, Coset, Cryptography, Cyclic group, D. C. Heath and Company, Data structure, Diffie–Hellman key exchange, Discrete logarithm, Disquisitiones Arithmeticae, Division (mathematics), Doomsday rule, Economics, Elliptic curve, Encryption, Encyclopædia Britannica, Enharmonic, Equal temperament, Equivalence class, Equivalence relation, Euclidean division, Euler's criterion, Euler's theorem, Euler's totient function, Exclusive or, Factorization of polynomials, Fermat's little theorem, ... Expand index (72 more) »

## Abstract algebra

In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures.

Addition (often signified by the plus symbol "+") is one of the four basic operations of arithmetic; the others are subtraction, multiplication and division.

The Advanced Encryption Standard (AES), also known by its original name Rijndael, is a specification for the encryption of electronic data established by the U.S. National Institute of Standards and Technology (NIST) in 2001.

## Apportionment (politics)

Apportionment is the process by which seats in a legislative body are distributed among administrative divisions entitled to representation.

## Arithmetic

Arithmetic (from the Greek ἀριθμός arithmos, "number") is a branch of mathematics that consists of the study of numbers, especially the properties of the traditional operations on them—addition, subtraction, multiplication and division.

## Binary relation

In mathematics, a binary relation on a set A is a set of ordered pairs of elements of A. In other words, it is a subset of the Cartesian product A2.

## Bitwise operation

In digital computer programming, a bitwise operation operates on one or more bit patterns or binary numerals at the level of their individual bits.

## Boolean ring

In mathematics, a Boolean ring R is a ring for which x2.

## Calculator

An electronic calculator is typically a portable electronic device used to perform calculations, ranging from basic arithmetic to complex mathematics.

## Carl Friedrich Gauss

Johann Carl Friedrich Gauss (Gauß; Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields, including algebra, analysis, astronomy, differential geometry, electrostatics, geodesy, geophysics, magnetic fields, matrix theory, mechanics, number theory, optics and statistics.

## Carmichael function

In number theory, the Carmichael function associates to every positive integer n a positive integer \lambda(n), defined as the smallest positive integer m such that (Dropping the phrase "between 1 and n" leads to an equivalent definition.) In algebraic terms, \lambda(n) equals the exponent of the multiplicative group of integers modulo ''n''.

## CAS Registry Number

A CAS Registry Number, also referred to as CASRN or CAS Number, is a unique numerical identifier assigned by the Chemical Abstracts Service (CAS) to every chemical substance described in the open scientific literature (currently including all substances described from 1957 through the present, plus some substances from the early or mid 1900s), including organic and inorganic compounds, minerals, isotopes, alloys and nonstructurable materials (UVCBs, of unknown, variable composition, or biological origin).

## Casting out nines

The expression "casting out nines" may refer to any one of three arithmetical procedures.

## Characteristic (algebra)

In mathematics, the characteristic of a ring R, often denoted char(R), is defined to be the smallest number of times one must use the ring's multiplicative identity (1) in a sum to get the additive identity (0) if the sum does indeed eventually attain 0.

## Charles E. Leiserson

Charles Eric Leiserson is a computer scientist, specializing in the theory of parallel computing and distributed computing, and particularly practical applications thereof.

## Check digit

A check digit is a form of redundancy check used for error detection on identification numbers, such as bank account numbers, which are used in an application where they will at least sometimes be input manually.

## Chemistry

Chemistry is the scientific discipline involved with compounds composed of atoms, i.e. elements, and molecules, i.e. combinations of atoms: their composition, structure, properties, behavior and the changes they undergo during a reaction with other compounds.

## Chinese remainder theorem

The Chinese remainder theorem is a theorem of number theory, which states that if one knows the remainders of the Euclidean division of an integer by several integers, then one can determine uniquely the remainder of the division of by the product of these integers, under the condition that the divisors are pairwise coprime.

## Circular buffer

A circular buffer, circular queue, cyclic buffer or ring buffer is a data structure that uses a single, fixed-size buffer as if it were connected end-to-end.

## Clifford Stein

Clifford Seth Stein (born December 14, 1965), a computer scientist, is a professor of industrial engineering and operations research at Columbia University in New York, NY, where he also holds an appointment in the Department of Computer Science.

## Commutative ring

In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative.

## Computer algebra

In computational mathematics, computer algebra, also called symbolic computation or algebraic computation, is a scientific area that refers to the study and development of algorithms and software for manipulating mathematical expressions and other mathematical objects.

## Computer science

Computer science deals with the theoretical foundations of information and computation, together with practical techniques for the implementation and application of these foundations.

## Congruence relation

In abstract algebra, a congruence relation (or simply congruence) is an equivalence relation on an algebraic structure (such as a group, ring, or vector space) that is compatible with the structure.

## Coprime integers

In number theory, two integers and are said to be relatively prime, mutually prime, or coprime (also written co-prime) if the only positive integer (factor) that divides both of them is 1.

## Coset

In mathematics, if G is a group, and H is a subgroup of G, and g is an element of G, then Only when H is normal will the set of right cosets and the set of left cosets of H coincide, which is one definition of normality of a subgroup.

## Cryptography

Cryptography or cryptology (from κρυπτός|translit.

## Cyclic group

In algebra, a cyclic group or monogenous group is a group that is generated by a single element.

## D. C. Heath and Company

D.C. Heath and Company was an American publishing company located at 125 Spring Street in Lexington, Massachusetts, specializing in textbooks.

## Data structure

In computer science, a data structure is a data organization and storage format that enables efficient access and modification.

## Diffie–Hellman key exchange

Diffie–Hellman key exchange (DH)Synonyms of Diffie–Hellman key exchange include.

## Discrete logarithm

In the mathematics of the real numbers, the logarithm logb a is a number x such that, for given numbers a and b. Analogously, in any group G, powers bk can be defined for all integers k, and the discrete logarithm logb a is an integer k such that.

## Disquisitiones Arithmeticae

The Disquisitiones Arithmeticae (Latin for "Arithmetical Investigations") is a textbook of number theory written in Latin by Carl Friedrich Gauss in 1798 when Gauss was 21 and first published in 1801 when he was 24.

## Division (mathematics)

Division is one of the four basic operations of arithmetic, the others being addition, subtraction, and multiplication.

## Doomsday rule

The Doomsday rule is an algorithm of determination of the day of the week for a given date.

## Economics

Economics is the social science that studies the production, distribution, and consumption of goods and services.

## Elliptic curve

In mathematics, an elliptic curve is a plane algebraic curve defined by an equation of the form which is non-singular; that is, the curve has no cusps or self-intersections.

## Encryption

In cryptography, encryption is the process of encoding a message or information in such a way that only authorized parties can access it and those who are not authorized cannot.

## Encyclopædia Britannica

The Encyclopædia Britannica (Latin for "British Encyclopaedia"), published by Encyclopædia Britannica, Inc., is a general knowledge English-language encyclopaedia.

## Enharmonic

In modern musical notation and tuning, an enharmonic equivalent is a note, interval, or key signature that is equivalent to some other note, interval, or key signature but "spelled", or named differently.

## Equal temperament

An equal temperament is a musical temperament, or a system of tuning, in which the frequency interval between every pair of adjacent notes has the same ratio.

## Equivalence class

In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation) defined on them, then one may naturally split the set S into equivalence classes.

## Equivalence relation

In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive.

## Euclidean division

In arithmetic, Euclidean division is the process of division of two integers, which produces a quotient and a remainder smaller than the divisor.

## Euler's criterion

In number theory Euler's criterion is a formula for determining whether an integer is a quadratic residue modulo a prime.

## Euler's theorem

In number theory, Euler's theorem (also known as the Fermat–Euler theorem or Euler's totient theorem) states that if n and a are coprime positive integers, then where \varphi(n) is Euler's totient function.

## Euler's totient function

In number theory, Euler's totient function counts the positive integers up to a given integer that are relatively prime to.

## Exclusive or

Exclusive or or exclusive disjunction is a logical operation that outputs true only when inputs differ (one is true, the other is false).

## Factorization of polynomials

In mathematics and computer algebra, factorization of polynomials or polynomial factorization is the process of expressing a polynomial with coefficients in a given field or in the integers as the product of irreducible factors with coefficients in the same domain.

## Fermat's little theorem

Fermat's little theorem states that if is a prime number, then for any integer, the number is an integer multiple of.

## Field (mathematics)

In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined, and behave as when they are applied to rational and real numbers.

## Finite field

In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements.

## Flat (music)

In music, flat or bemolle (Italian: "soft B") means "lower in pitch".

## Floating-point arithmetic

In computing, floating-point arithmetic is arithmetic using formulaic representation of real numbers as an approximation so as to support a trade-off between range and precision.

## Game theory

Game theory is "the study of mathematical models of conflict and cooperation between intelligent rational decision-makers".

## Gaussian elimination

In linear algebra, Gaussian elimination (also known as row reduction) is an algorithm for solving systems of linear equations.

## Gröbner basis

In mathematics, and more specifically in computer algebra, computational algebraic geometry, and computational commutative algebra, a Gröbner basis is a particular kind of generating set of an ideal in a polynomial ring over a field.

## Group theory

In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.

## Ideal (ring theory)

In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring.

## Integer

An integer (from the Latin ''integer'' meaning "whole")Integer&#x2009;'s first literal meaning in Latin is "untouched", from in ("not") plus tangere ("to touch").

## Integer factorization

In number theory, integer factorization is the decomposition of a composite number into a product of smaller integers.

## International Bank Account Number

The International Bank Account Number (IBAN) is an internationally agreed system of identifying bank accounts across national borders to facilitate the communication and processing of cross border transactions with a reduced risk of transcription errors.

## International Data Encryption Algorithm

In cryptography, the International Data Encryption Algorithm (IDEA), originally called Improved Proposed Encryption Standard (IPES), is a symmetric-key block cipher designed by James Massey of ETH Zurich and Xuejia Lai and was first described in 1991.

## International Standard Book Number

The International Standard Book Number (ISBN) is a unique numeric commercial book identifier.

## Introduction to Algorithms

Introduction to Algorithms is a book by Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein.

## Isomorphism

In mathematics, an isomorphism (from the Ancient Greek: ἴσος isos "equal", and μορφή morphe "form" or "shape") is a homomorphism or morphism (i.e. a mathematical mapping) that can be reversed by an inverse morphism.

## Knot theory

In topology, knot theory is the study of mathematical knots.

## Lagrange's theorem (group theory)

Lagrange's theorem, in the mathematics of group theory, states that for any finite group G, the order (number of elements) of every subgroup H of G divides the order of G. The theorem is named after Joseph-Louis Lagrange.

## Lagrange's theorem (number theory)

In number theory, Lagrange's theorem is a statement named after Joseph-Louis Lagrange about how frequently a polynomial over the integers may evaluate to a multiple of a fixed prime.

## Law

Law is a system of rules that are created and enforced through social or governmental institutions to regulate behavior.

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## Linear algebra

Linear algebra is the branch of mathematics concerning linear equations such as linear functions such as and their representations through matrices and vector spaces.

## Long double

In C and related programming languages, long double refers to a floating-point data type that is often more precise than double-precision.

## Mathematics

Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.

## Maximal ideal

In mathematics, more specifically in ring theory, a maximal ideal is an ideal that is maximal (with respect to set inclusion) amongst all proper ideals.

## Modular exponentiation

Modular exponentiation is a type of exponentiation performed over a modulus.

## Modular multiplicative inverse

In mathematics, in particular the area of number theory, a modular multiplicative inverse of an integer is an integer such that the product is congruent to 1 with respect to the modulus.

## Modulo operation

In computing, the modulo operation finds the remainder after division of one number by another (sometimes called modulus).

## Montgomery modular multiplication

In modular arithmetic computation, Montgomery modular multiplication, more commonly referred to as Montgomery multiplication, is a method for performing fast modular multiplication.

## Multiple (mathematics)

In science, a multiple is the product of any quantity and an integer.

## Multiplication

Multiplication (often denoted by the cross symbol "×", by a point "⋅", by juxtaposition, or, on computers, by an asterisk "∗") is one of the four elementary mathematical operations of arithmetic; with the others being addition, subtraction and division.

## Multiplicative group of integers modulo n

In modular arithmetic, the integers coprime (relatively prime) to n from the set \ of n non-negative integers form a group under multiplication modulo n, called the multiplicative group of integers modulo n. Equivalently, the elements of this group can be thought of as the congruence classes, also known as residues modulo n, that are coprime to n. Hence another name is the group of primitive residue classes modulo n. In the theory of rings, a branch of abstract algebra, it is described as the group of units of the ring of integers modulo n. Here units refers to elements with a multiplicative inverse, which in this ring are exactly those coprime to n. This group, usually denoted (\mathbb/n\mathbb)^\times, is fundamental in number theory.

## Music

Music is an art form and cultural activity whose medium is sound organized in time.

## NP-completeness

In computational complexity theory, an NP-complete decision problem is one belonging to both the NP and the NP-hard complexity classes.

## NP-intermediate

In computational complexity, problems that are in the complexity class NP but are neither in the class P nor NP-complete are called NP-intermediate, and the class of such problems is called NPI.

## Number theory

Number theory, or in older usage arithmetic, is a branch of pure mathematics devoted primarily to the study of the integers.

## Octave

In music, an octave (octavus: eighth) or perfect octave is the interval between one musical pitch and another with half or double its frequency.

## Pisano period

In number theory, the nth Pisano period, written π(n), is the period with which the sequence of Fibonacci numbers taken modulo n repeats.

## Polynomial

In mathematics, a polynomial is an expression consisting of variables (also called indeterminates) and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables.

## Polynomial greatest common divisor

In algebra, the greatest common divisor (frequently abbreviated as GCD) of two polynomials is a polynomial, of the highest possible degree, that is a factor of both the two original polynomials.

## Prentice Hall

Prentice Hall is a major educational publisher owned by Pearson plc.

## Prime number

A prime number (or a prime) is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers.

## Prime power

In mathematics, a prime power is a positive integer power of a single prime number.

## Primitive root modulo n

In modular arithmetic, a branch of number theory, a number g is a primitive root modulo n if every number a coprime to n is congruent to a power of g modulo n. That is, for every integer a coprime to n, there is an integer k such that gk ≡ a (mod n).

## Programming language

A programming language is a formal language that specifies a set of instructions that can be used to produce various kinds of output.

## Proportional division

A proportional division is a kind of fair division in which a resource is divided among n partners with subjective valuations, giving each partner at least 1/n of the resource by his/her own subjective valuation.

## Public-key cryptography

Public-key cryptography, or asymmetric cryptography, is any cryptographic system that uses pairs of keys: public keys which may be disseminated widely, and private keys which are known only to the owner.

In number theory, the law of quadratic reciprocity is a theorem about modular arithmetic that gives conditions for the solvability of quadratic equations modulo prime numbers.

In number theory, an integer q is called a quadratic residue modulo n if it is congruent to a perfect square modulo n; i.e., if there exists an integer x such that: Otherwise, q is called a quadratic nonresidue modulo n. Originally an abstract mathematical concept from the branch of number theory known as modular arithmetic, quadratic residues are now used in applications ranging from acoustical engineering to cryptography and the factoring of large numbers.

## Quotient group

A quotient group or factor group is a mathematical group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves the group structure.

## Quotient ring

In ring theory, a branch of abstract algebra, a quotient ring, also known as factor ring, difference ring or residue class ring, is a construction quite similar to the quotient groups of group theory and the quotient spaces of linear algebra.

## Rational reconstruction (mathematics)

In mathematics, rational reconstruction is a method that allows one to recover a rational number from its value modulo an integer.

## RC4

In cryptography, RC4 (Rivest Cipher 4 also known as ARC4 or ARCFOUR meaning Alleged RC4, see below) is a stream cipher.

## Reduced residue system

Any subset R of the integers is called a reduced residue system modulo n if.

## Ring (mathematics)

In mathematics, a ring is one of the fundamental algebraic structures used in abstract algebra.

## Ring theory

In algebra, ring theory is the study of rings—algebraic structures in which addition and multiplication are defined and have similar properties to those operations defined for the integers.

## Ron Rivest

Ronald Linn Rivest (born May 6, 1947) is a cryptographer and an Institute Professor at MIT.

## RSA (cryptosystem)

RSA (Rivest–Shamir–Adleman) is one of the first public-key cryptosystems and is widely used for secure data transmission.

## Serial number arithmetic

Many protocols and algorithms require the serialization or enumeration of related entities.

## Sharp (music)

In music, sharp, dièse (from French), or diesis (from Greek) means higher in pitch.

## Singleton (mathematics)

In mathematics, a singleton, also known as a unit set, is a set with exactly one element.

## Social science

Social science is a major category of academic disciplines, concerned with society and the relationships among individuals within a society.

## Subtraction

Subtraction is an arithmetic operation that represents the operation of removing objects from a collection.

## Symmetric-key algorithm

Symmetric-key algorithms are algorithms for cryptography that use the same cryptographic keys for both encryption of plaintext and decryption of ciphertext.

## Thomas H. Cormen

Thomas H. Cormen is the co-author of Introduction to Algorithms, along with Charles Leiserson, Ron Rivest, and Cliff Stein.

## Thue's lemma

In modular arithmetic, Thue's lemma roughly states that every modular integer may be represented by a "modular fraction" such that the numerator and the denominator have absolute values not greater than the square root of the modulus.

## Time complexity

In computer science, the time complexity is the computational complexity that describes the amount of time it takes to run an algorithm.

## Two-element Boolean algebra

In mathematics and abstract algebra, the two-element Boolean algebra is the Boolean algebra whose underlying set (or universe or carrier) B is the Boolean domain.

## Visual arts

The visual arts are art forms such as ceramics, drawing, painting, sculpture, printmaking, design, crafts, photography, video, filmmaking, and architecture.

## Wilson's theorem

In number theory, Wilson's theorem states that a natural number n > 1 is a prime number if and only if the product of all the positive integers less than n is one less than a multiple of n. That is (using the notations of modular arithmetic), one has that the factorial (n - 1)!.

## Zeller's congruence

Zeller's congruence is an algorithm devised by Christian Zeller to calculate the day of the week for any Julian or Gregorian calendar date.

## 12-hour clock

The 12-hour clock is a time convention in which the 24 hours of the day are divided into two periods: "The use of AM or PM to designate either noon or midnight can cause ambiguity.

## References

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